Chapter 6 - Knot Diagrammatics

Abstract

This chapter reviews knot theory and explains the invariants of knots and links from the point of view of categories of diagrams. It discusses the foundations of knot theory, focusing on virtual knot theory and topological quantum field theory. The chapter also presents a discussion on graph embeddings that extends Reidemeister's theorem to graphs and proves the appropriate moves for topological and rigid vertices. Discussions of the four-term relations, Lie algebra weights, relationships with the Witten functional intergal, and combinatorial constructions for some Vassiliev invariants are also presented. Reidemeister's theorem states that two diagrams represent ambient isotopic knots (or links) if and only if there is a sequence of Reidemeister moves taking one diagram to the other. Diagrams are always subject to topological deformations in the plane that preserve the structure of the crossings. These deformations could be designated as “Move Zero”.